Found problems: 85335
Let $x,y,z$ be complex numbers such that\\
$\hspace{ 2cm} \frac{x}{y+z}+\frac{y}{z+x}+\frac{z}{x+y}=9$\\
$\hspace{ 2cm} \frac{x^2}{y+z}+\frac{y^2}{z+x}+\frac{z^2}{x+y}=64$\\
$\hspace{ 2cm} \frac{x^3}{y+z}+\frac{y^3}{z+x}+\frac{z^3}{x+y}=488$\\
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If $\frac{x}{yz}+\frac{y}{zx}+\frac{z}{xy}=\frac{m}{n}$ where $m,n$ are positive integers with $GCD(m,n)=1$, find $m+n$.
Let $ABC$ be a triangle with $AB = 4024$, $AC = 4024$, and $BC=2012$. The reflection of line $AC$ over line $AB$ meets the circumcircle of $\triangle{ABC}$ at a point $D\ne A$. Find the length of segment $CD$.
[i]Ray Li.[/i]
Point $D$ lies on side $AB$ of triangle $ABC$, and $$\frac{AD}{DC} = \frac{AB}{BC}.$$
Prove that angle $C$ is obtuse.
(Sergey Berlov)
We label every edge of a simple graph with the difference of the degrees of its endpoints. If the number of vertices is $N$, what can be the largest value of the sum of the labels on the edges?
[i]Proposed by Dániel Lenger and Gábor Szűcs, Budapest[/i]
Consider the ten numbers $ar, ar^2, ar^3, ... , ar^{10}$. If their sum is 18 and the sum of their reciprocals is 6, determine their product.
2) A mass $m$ hangs from a massless spring connected to the roof of a box of mass $M$. When the box is held stationary, the mass–spring system oscillates vertically with angular frequency $\omega$. If the box is dropped and falls freely under gravity, how will the angular frequency change?
A) $\omega$ will be unchanged
B) $\omega$ will increase
C) $\omega$ will decrease
D) Oscillations are impossible under these conditions.
E) $\omega$ will either increase or decrease depending on the values of $M$ and $m$.
Let $ \triangle ABE $ be a triangle with $ \frac{AB}{3} = \frac{BE}{4} = \frac{EA}{5} $. Let $ D \neq A $ be on line $ \overline{AE} $ such that $ AE = ED $ and $ D $ is closer to $ E $ than to $ A $. Moreover, let $ C $ be a point such that $ BCDE $ is a parallelogram. Furthermore, let $ M $ be on line $ \overline{CD} $ such that $ \overline{AM} $ bisects $ \angle BAE $, and let $ P $ be the intersection of $ \overline{AM} $ and $ \overline{BE} $. Compute the ratio of $ PM $ to the perimeter of $ \triangle ABE $.
Find all natural numbers $n \ge 3$ for which in an arbitrary $n$-gon one can choose $3$ vertices dividing its boundary into three parts, the lengths of which can be the lengths of the sides of some triangle.
(Fedir Yudin)
The number $28!$ ($28$ in decimal) has base $30$ representation \[28!=Q6T32S??OCLQJ6000000_{30}\] where the seventh and eighth digits are missing. What are the missing digits? In base $30$, we have that the digits $A=10$, $B=11$, $C=12$, $D=13$, $E=14$, $F=15$, $G=16$, $H=17$, $I=18$, $J=19$, $K=20$, $L=21$, $M=22$, $N=23$, $O=24$, $P=25$, $Q=26$, $R=27$, $S=28$, $T=29$.
[i]2019 CCA Math Bonanza Tiebreaker Round #4[/i]
In trapezoid $ABCD$, we have $\overline{AD}\parallel\overline{BC}$, $BC=3$, and $CD=4$. In addition, $\cos\angle ADC=\tfrac{1}{3}$ and $\angle ABC=2\angle ADC$. Compute $AC$.
Determine all integers $a$ such that $a^k + 1$ is divisible by $12321$ for some $k$
Anna and Basilis play a game writing numbers on a board as follows:
The two players play in turns and if in the board is written the positive integer $n$, the player whose turn is chooses a prime divisor $p$ of $n$ and writes the numbers $n+p$. In the board, is written at the start number $2$ and Anna plays first. The game is won by whom who shall be first able to write a number bigger or equal to $31$.
Find who player has a winning strategy, that is who may writing the appropriate numbers may win the game no matter how the other player plays.
How many ordered pairs of nonnegative integers $\left(a,b\right)$ are there with $a+b=999$ such that each of $a$ and $b$ consists of at most two different digits? (These distinct digits need not be the same digits in both $a$ and $b$. For example, we might have $a=622$ and $b=377$.)
Find all solutions (real and complex) for $x,y,z$, given that:
\[ x+y+z = 6 \\
x^2+y^2+z^2 = 12 \\
x^3+y^3+z^3 = 24 \]
Let $N = \frac{1}{3} + \frac{3}{5} + \frac{5}{7} + \frac{7}{9} + \frac{9}{11}$. What is the greatest integer which is less than $N$?
$(MOR 1)$ Denote by $[x]$ the greatest integer not exceeding $x$. For all real $k > 1$, define two sequences:
\[a_n(k) = [nk]\text{ and } b_n(k) =\left[\frac{nk}{k - 1}\right]\]
If $A(k) = \{a_n(k) : n \in\mathbb{N}\}$ and $B(k) = \{b_n(k) : n \in \mathbb{N}\}$, prove that $A(k)$ and $B(k)$ form a partition of $\mathbb{N}$ if and only if $k$ is irrational.
All possible $6$-digit numbers, in each of which the digits occur in nonincreasing order (from left to right, e.g. $877550$) are written as a sequence in increasing order. Find the $2005$-th number in this sequence.
Alice, Bob, and Conway are playing rock-paper-scissors. Each player plays against each of the other $2$ players and each pair plays until a winner is decided (i.e. in the event of a tie, they play again). What is the probability that each player wins exactly once?
Let $k,l$ be two given integers. Prove that there exist infinite many integers $m\ge k$ such that $\gcd\left(\binom{m}{k},l\right)=1$.
Ten Cs are written in a row. Some Cs are upper-case and some are lower-case, and each is written in one of two colors, green and yellow. It is given that there is at least one lower-case C, at least one green C, and at least one C that is both upper-case and yellow. Furthermore, no lower-case C can be followed by an upper-case C, and no yellow C can be followed by a green C. In how many ways can the Cs be written?
The number $1$ is written on the blackboard. At any point, Kornny may pick two (not necessary distinct) of the numbers $a$ and $b$ written on the board and write either $ab$ or $\frac{1}{a}+\frac{1}{b}+\frac{1}{ab}$ on the board as well. Determine all possible numbers that Kornny can write on the board in finitely many steps.
There are 4 countries: Argentina, Brazil, Peru and Uruguay. Each country consists of 4 islands. There are bridges going back and forth between some of the 16 islands. Carlos noted that whenever he travels between some of the islands using the bridges, without using the same bridge twice, and ending in the island where he started his journey, he will necessarily visit at least one island of each country.
Determine the maximum number of bridges there can be.
$ f(x)$ and $ g(x)$ are two polynomials with nonzero degrees and integer coefficients, such that $ g(x)$ is a divisor of $ f(x)$ and the polynomial $ f(x)\plus{}2009$ has $ 50$ integer roots. Prove that the degree of $ g(x)$ is at least $ 5$.
Prove that for any triangle each exradius is less than four times the circumradius.
In the acute triangle $ABC$, the sum of the distances from the vertices $B$ and $C$ to of the orthocenter $H$ is equal to $4r,$ where $r$ is the radius of the circle inscribed in this triangle. Find the perimeter of triangle $ABC$ if it is known that $BC=a$.
(Gryhoriy Filippovskyi)